Math 9 September 29, 21 Topics in Applied Mathematics: Introduction to the Mathematics of Climate Mondays and Wednesdays 2: : http://www.math.umn.edu/~mcgehee/teaching/math9-21-2fall/ Streaming video is available at http://www.ima.umn.edu/videos/ Click on the link: "Live Streaming from Lind Hall". Participation: https://umconnect.umn.edu/mathclimate The Big Picture http://www.snowballearth.org/when.html Temperatures in the Cenozoic Era 18 O in Foraminifera Fossils During the Past. Myr Benthic Data (δ18o)... 2 2 1 1 Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J. 2 (28) Lisiecki, L. E., and M. E. Raymo (2), A Pliocene-Pleistocene stack of 7 globally distributed benthic d18o records, Paleoceanography,2, PA1, doi:1.129/2pa171. 18 O in Foraminifera Fossils During the Past 1. Myr Recent (last ) Temperature Cycles Vostok Ice Core Data Benthic Data (δ18o)... 1 9 8 7 6 2 1 Lisiecki, L. E., and M. E. Raymo (2), A Pliocene-Pleistocene stack of 7 globally distributed benthic d18o records, Paleoceanography,2, PA1, doi:1.129/2pa171. J.R. Petit, et al (1999) Climate and atmospheric history of the past 2, years from the Vostok ice core, Antarctica, Nature 99, 29-6. Richard McGehee, University of Minnesota 1
Eccentricity Eccentricity....1. 1 9 8 7 6 2 1 Note periods of about 1 kyr and kyr. John Imbrie & Katherine Palmer Imbrie, Ice Ages: Solving the Mystery, Harvard Univ. Press, 1979. The effect due to is more significant, but not that much: As e varies between and, (1-e 2 ) -1/2 varies between 1 and 1.18, or about.2%. (Twenty times the effect due to a.) J. Laskar, et al (2) A long-term numerical solution for the insolation quantities of the Earth, Astronomy & Astrophysics 28, 261 28. Obliquity Obliquity 2. 2. (degrees) 2. 2. 2 22. 1 9 8 7 6 2 1 Note period of about 1. http://upload.wikimedia.org/wikipedia/commons/6/61/axialtiltobliquity.png J. Laskar, et al (2) A long-term numerical solution for the insolation quantities of the Earth, Astronomy & Astrophysics 28, 261 28. Precession Precession Index... 1 9 8 7 6 2 1 http://earthobservatory.nasa.gov/library/giants/milankovitch/milankovitch_2.html Math 9 9//21 index = e sinρ, where e = and ρ = angle (measured from spring equinox) Note period of about 2. J. Laskar, et al (2) A long-term numerical solution for the insolation quantities of the Earth, Astronomy & Astrophysics 28, 261 28. Math 9 9//21 Richard McGehee, University of Minnesota 2
Precession Daily Average Insolation at Summer Solstice at 6 N 8 6 2 W/m^2 8 6 2 1 9 8 7 6 2 1 http://en.wikipedia.org/wiki/zodiac Who was Milankovitch? Milutin Milankovitch was a Serbian mathematician and professor at the University of Belgrade. In 192 he published his seminal work on the relation between insolation and the Earth s orbital parameters. Milutin Milankovitch 1879-198 In 191 he published a book explaining his entire theory. http://en.wikipedia.org/wiki/milankovitch_cycles His work was not fully accepted until 1976. What happened in 1976? Suggested Reading Hays, Imbrie, and Shackleton, Variations in the Earth's Orbit: Pacemaker of the Ice Ages, Science 19, 1 December 1976. James D. Hays John Imbrie It is concluded that changes in the earth's orbital geometry are the fundamental cause of the succession of Quaternary ice ages. Nicholas Shackleton John Imbrie & Katherine Palmer Imbrie, Ice Ages: Solving the Mystery, HARVARD UNIVERSITY PRESS, 1979 Richard McGehee, University of Minnesota
History of Discovery Agassiz announces glacial theory Evidence of multiple ice ages discovered in Illinois Magnetic reversals discovered 18 O theory developed Fourier climate fluctuations found in ocean cores paleomagnetic time scale developed Hays, et al 18 182 18 186 188 19 192 19 196 198 2 Milankovitch explains glacial cycles Croll explains glacial cycles Humboldt debunks Adhemar Adhemar explains glacial cycles Richard McGehee, University of Minnesota
Suggested Reading Milankovitch Theory, verified by Hays, et al: The glacial cycles are driven by the variations in the Earth s orbit (Milankovitch Cycles), causing a variation in incoming solar radiation (insolation). We see the evidence for the existence of glacial cycles in the climate record. We can compute the Milankovitch cycles from classical physics and evidence has been gathered for thousands of years. What is the evidence for the connection between glacial cycles and Milankovitch cycles? Dan Brown, The Da Vinci Code, ANCHOR BOOKS, 2 Example: The ice ages occur about every 1, year, and the of the Earth s orbit cycles through changes every 1, years. Can we quantify this observation? Example: The ice ages occur about every 1, year, and the of the Earth s orbit cycles through changes every 1, years. Can we quantify this observation? Fourier Transform (Power Spectrum) time series spectrum Solar Forcing (Hays, et al) Climate Response, Hays, et al Three different temperature proxies from sea sediment data. Hays, et al, Science 19 (1976), p. 112 Hays, et al, Science 19 (1976), p. 112 Richard McGehee, University of Minnesota
Eccentricity Obliquity....1 2 2 1 1 (degrees) 2. 2. 2. 2. 2 22. 2 2 1 1 Precession Index Spectral Analysis of the Milankovitch cycles. index.. Laskar s computations....1 2 2 1 1 Eccentricity Spectra 2 2 1 1 2. (degrees) 2. 2. 2. 2 Obliquity 22. 2 2 1 1 index.. Precession 2 18 16 1 12 1 8 6 2 Summer Solstice 6 N Climate Response, Hays, et al 8 W/m 2 6 2 2 2 1 1? Three different temperature proxies from sea sediment data. Hays, et al, Science 19 (1976), p. 112 Richard McGehee, University of Minnesota 6
Hays, et al, Summary Hays with Modern Data 2)... climatic variance of these records is concentrated in three discrete spectral peaks at periods of 2,, 2,, and approximately 1, years. These peaks correspond to the dominant periods of the earth's solar orbit, and contain respectively about 1, 2, and percent of the climatic variance. d18o... 2 2 1 1 Hays, et al, Science 19 (1976), p. 112 Milankovitch vs. Climate Hays, et al, Summary Forcing Response? Increasing contribution Hays explanation is that there are nonlinear feedbacks. Are there other explanations? Hays, et al, Science 19 (1976), p. 1127 Cenozoic Era Temperatures in the Cenozoic Era Zachos, et al, Science 292 (21), p. 689 Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J. 2 (28) Richard McGehee, University of Minnesota 7
Climate Response (Zachos, et al) Zachos, et al, Summary Forcing Response Power spectrum of climate for the last. Myr. Note the peaks at 1 and 1. Increasing contribution Nonlinear effects? Other explanations? Zachos, et al, Science 292 (21), p. 689 Zachos, et al, Science 292 (21), p. 689 Spectral Analysis of the Climate Data Spectral Analysis of the Climate Data... δ 18 O..... 2 2 1 1 1 1 2 δ 18 O.... 2 2 1 1 1 1 2 Conclusion Remains: The Milankovitch cycles pace the Earth s climate. Exactly how is not so clear...1... frequency (1/)..1... frequency (1/) Why such a small contribution? Zachos Summary (Revised) Incoming Solar Radiation (Insolation), averaged over the entire globe and over a full year, depends only on e, not on either or. Q Q e 2 1 e Insolation as a function of latitude, averaged over a full year, depends on e and β, but not. I Q e s y, where 2 2 2 s, 1 2 cos sin cos sin cos cos d latitude If we assume that glaciation depends on annual average insolation instead of insolation at summer solstice, then forcing and response are aligned. Increasing contribution Forcing Response Richard McGehee, University of Minnesota 8
Something s Missing Something s Missing 2. Lisiecki Raymo Stack (degrees) 2. 2. 2. 2 22. 2 2 1 1 Lisiecki Raymo Stack data d18o... 6 2 1 1 1 2 climate d18o.. spectrum?. 6 2 1..1... frequency (1/) A transition occurred about one million years ago: the amplitude increased and the dominant period changed from 1 kyr to 1 kyr. - to Myr - to -1 Myr -1 to Myr Last Million Years is Different 1 1 2?..1... frequency (1/) data What s up with the Last Million Years? 1, Year Problem: Why did the signal become so dominant during the last million years?, Year Problem: If the last million years is dominated by, what happened to the, year cycle? 1 1 2?..1... frequency (1/) Richard McGehee, University of Minnesota 9